多变量线性回归featureNormalize gradientDescentMulti normalEqn computeCostMulti
OptionalExercises
1.多变量线性回归
本部分使用多变量线性回归来预测房价。假设你想卖房子并想知道一个合适的市场价应该是多少。一种方式就是如这样收集最近的房价并做一个房价模型。Ex1data2.txt包含了Portland、Oregon等地房价作为训练集。第一列是房子尺寸（平方英尺），第二列是卧室的数量，第三列是房子的价格。
Ex1_multi.m已经为你准备好了来完成这个练习。
1.1FeatureNormalization
ex1_multi.m一开始会加载并显示一些数据集中的值。通过看这些值，注意到房价是卧室数的近1000倍。当特征之间的量级不一样，第一个特征的规模能够使梯度下降收敛的更快。
你的任务是完成featureNormalize.m中的code来：
A. 数据集中的每个特征值减去他们的平均值。
B. 减后，用这些值除他们各自的标准差。
标准差是一种表示一个特征的波动范围的测量方式；这是一种可供选择的计算值的范围（max-min）。Octave/matlab中你可以使用‘std’命令来计算标准差。例如，featureNormalize.m中X(:,1)的值包含所有的x1（房价）的值，所以std(X(:,1))计算了所以房价的标准差。此时，额外的列x0=1还没有加入到X中（详细看ex1_multi.m）。
你会为所有特征都执行这种操作，因此你的code应该可用于所有尺寸的数据集。注意到矩阵X的每一列对应一个特征。
function [X_norm, mu, sigma] = featureNormalize(X)
%FEATURENORMALIZE Normalizes the features in X
% FEATURENORMALIZE(X) returns a normalized version of X where
% the mean value of each feature is 0 and the standard deviation
% is 1. This is often a good preprocessing step to do when
% working with learning algorithms.
% You need to set these values correctly
X_norm = X;
mu = zeros(1, size(X, 2));
sigma = zeros(1, size(X, 2));
% ====================== YOUR CODE HERE ======================
% Instructions: First, for each feature dimension, compute the mean
% of the feature and subtract it from the dataset,
% storing the mean value in mu. Next, compute the
% standard deviation of each feature and divide
% each feature by it's standard deviation, storing
% the standard deviation in sigma.
%
% Note that X is a matrix where each column is a
% feature and each row is an example. You need
% to perform the normalization separately for
% each feature.
%
% Hint: You might find the 'mean' and 'std' functions useful.
%
mu=mean(X); % 1 x n;
sigma=std(X); % 1 x n;
mu_temp=((mu')*ones(1,m))'; % m x n
sigma_temp=(sigma'*ones(1,m))' ; % m x n
X_norm=(X-mu_temp)./sigma_temp;
% ============================================================
end
1.2GradientDescent
在之前已经在单变量回归问题使用了梯度下降。不同之处是，这里多了一个变量。假设函数和批量梯度下降更新准则没有变。需要完成computeCostMulti.m和gradientDescentMulti.m中的代码用于多变量线性回归。确保你的代码支撑任意特征值并很好的向量化了。可以使用‘size(X,2)’来看数据中有多少个特征。
function [theta, J_history] = gradientDescentMulti(X, y, theta, alpha, num_iters)
%GRADIENTDESCENTMULTI Performs gradient descent to learn theta
% theta = GRADIENTDESCENTMULTI(x, y, theta, alpha, num_iters) updates theta by
% taking num_iters gradient steps with learning rate alpha
% Initialize some useful values
m = length(y); % number of training examples
J_history = zeros(num_iters, 1);
for iter = 1:num_iters
% ====================== YOUR CODE HERE ======================
% Instructions: Perform a single gradient step on the parameter vector
% theta.
%
% Hint: While debugging, it can be useful to print out the values
% of the cost function (computeCostMulti) and gradient here.
%
predictions = X*theta;
errors = (predictions-y);
sums = X'*errors;
delta = 1/m*sums;
theta = theta - alpha*delta;
% ============================================================
% Save the cost J in every iteration
J_history(iter) = computeCostMulti(X, y, theta);
end
end
function J = computeCostMulti(X, y, theta)
%COMPUTECOSTMULTI Compute cost for linear regression with multiple variables
% J = COMPUTECOSTMULTI(X, y, theta) computes the cost of using theta as the
% parameter for linear regression to fit the data points in X and y
% Initialize some useful values
m = length(y); % number of training examples
% You need to return the following variables correctly
J = 0;
% ====================== YOUR CODE HERE ======================
% Instructions: Compute the cost of a particular choice of theta
% You should set J to the cost.
m = size(X,1);
predictions = X*theta;
errors = predictions-y;
J = 1/(2*m)*(errors)'*errors;
% =========================================================================
end注意，在多变量情况下，cost函数也能被写为以下向量格式：
1.2.1Optional（ungraded）exercise：selecting learning raets
本部分中，对于数据集使用不同的学习速率并寻找最快收敛的学习速率。通过修改ex1_multi.m和改变部分关于学习速率的代码来改变学习速率。
ex1_multi.m中的下一部分会调用你的gradientDescentMulti.m函数，并在选择的学习速率下迭代50次。函数还会在向量J中返回J(theta)的历史值。最后一次迭代之后，画出J(theta)相对于迭代次数的J的值。如果你选择一个好的学习速率，你画出来的应类似图4.如果你的图看起来非常不同，尤其是如果你的J(theta)的值增加或者波动，修改你的学习速率并再次尝试。我们推荐以log的规模，3倍增加的形式来选学习速率（如0.3，0.1，0.03,0.01）。
1.3normalEquations
在课程视频中， 学习到了和线性回归相似的解法是使用这个公式，不需要任何特征scaling，并且你会在一次计算中得到一个额外的解：这里也没有像梯度下降中的“循环直到收敛”。完成normalEqn.m中的代码使用上述的公式计算θ。记得你不用缩放你的特征，我们依旧需要加一个全1列在X中，来获得截断项（theta_0）。
function [theta] = normalEqn(X, y)
%NORMALEQN Computes the closed-form solution to linear regression
% NORMALEQN(X,y) computes the closed-form solution to linear
% regression using the normal equations.
theta = zeros(size(X, 2), 1);
% ====================== YOUR CODE HERE ======================
% Instructions: Complete the code to compute the closed form solution
% to linear regression and put the result in theta.
%
% ---------------------- Sample Solution ----------------------
theta = pinv( X' * X ) * X' * y;
% ------------------------------------------------------------
% ============================================================
end
%% Machine Learning Online Class
% Exercise 1: Linear regression with multiple variables
%
% Instructions
% ------------
% This file contains code that helps you get started on the
% linear regression exercise.
%
% You will need to complete the following functions in this
% exericse:
%
% warmUpExercise.m
% plotData.m
% gradientDescent.m
% computeCost.m
% gradientDescentMulti.m
% computeCostMulti.m
% featureNormalize.m
% normalEqn.m
%
% For this part of the exercise, you will need to change some
% parts of the code below for various experiments (e.g., changing
% learning rates).
%
%% Initialization
%% ================ Part 1: Feature Normalization ================
%% Clear and Close Figures
clear ; close all; clc
fprintf('Loading data ...\n');
%% Load Data
data = load('ex1data2.txt');
X = data(:, 1:2);
y = data(:, 3);
m = length(y);
% Print out some data points
fprintf('First 10 examples from the dataset: \n');
fprintf(' x = [%.0f %.0f], y = %.0f \n', [X(1:10,:) y(1:10,:)]');
fprintf('Program paused. Press enter to continue.\n');
pause;
% Scale features and set them to zero mean
fprintf('Normalizing Features ...\n');
[X mu sigma] = featureNormalize(X);
% Add intercept term to X
X = [ones(m, 1) X];
%% ================ Part 2: Gradient Descent ================
% ====================== YOUR CODE HERE ======================
% Instructions: We have provided you with the following starter
% code that runs gradient descent with a particular
% learning rate (alpha).
%
% Your task is to first make sure that your functions -
% computeCost and gradientDescent already work with
% this starter code and support multiple variables.
%
% After that, try running gradient descent with
% different values of alpha and see which one gives
% you the best result.
%
% Finally, you should complete the code at the end
% to predict the price of a 1650 sq-ft, 3 br house.
%
% Hint: By using the 'hold on' command, you can plot multiple
% graphs on the same figure.
%
% Hint: At prediction, make sure you do the same feature normalization.
%
fprintf('Running gradient descent ...\n');
% Choose some alpha value
alpha = 0.01;
num_iters = 400;
% Init Theta and Run Gradient Descent
theta = zeros(3, 1);
[theta, J_history] = gradientDescentMulti(X, y, theta, alpha, num_iters);
% Plot the convergence graph
figure;
plot(1:numel(J_history), J_history, '-b', 'LineWidth', 2);
xlabel('Number of iterations');
ylabel('Cost J');
% Display gradient descent's result
fprintf('Theta computed from gradient descent: \n');
fprintf(' %f \n', theta);
fprintf('\n');
% Estimate the price of a 1650 sq-ft, 3 br house
% ====================== YOUR CODE HERE ======================
% Recall that the first column of X is all-ones. Thus, it does
% not need to be normalized.
% price = 0; % You should change this
price = [1 (([1650 3]-mu) ./ sigma)] * theta ;
% ============================================================
fprintf(['Predicted price of a 1650 sq-ft, 3 br house ' ...
'(using gradient descent):\n $%f\n'], price);
fprintf('Program paused. Press enter to continue.\n');
pause;
%% ================ Part 3: Normal Equations ================
fprintf('Solving with normal equations...\n');
% ====================== YOUR CODE HERE ======================
% Instructions: The following code computes the closed form
% solution for linear regression using the normal
% equations. You should complete the code in
% normalEqn.m
%
% After doing so, you should complete this code
% to predict the price of a 1650 sq-ft, 3 br house.
%
%% Load Data
data = csvread('ex1data2.txt');
X = data(:, 1:2);
y = data(:, 3);
m = length(y);
% Add intercept term to X
X = [ones(m, 1) X];
% Calculate the parameters from the normal equation
theta = normalEqn(X, y);
% Display normal equation's result
fprintf('Theta computed from the normal equations: \n');
fprintf(' %f \n', theta);
fprintf('\n');
% Estimate the price of a 1650 sq-ft, 3 br house
% ====================== YOUR CODE HERE ======================
% price = 0; % You should change this
price = [1 1650 3] * theta ;
% ============================================================
fprintf(['Predicted price of a 1650 sq-ft, 3 br house ' ...
'(using normal equations):\n $%f\n'], price);
-----------------
作者：思念铭心
来源：CSDN
原文：https://blog.csdn.net/sinat_38930882/article/details/77934734
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